In addition to contribution 3. & 4., the results establish the decay of the mean square error (MSE), for instance, of the log-normalizing constant estimate as the time
parameter increases. This rate is expected to occur in practice, as we will see in simulations, and parallels other results found in the literature for particle
estimates of the normalization constant (e.g. such as particle filters [1]}). In relation to 5., if one assumes that the model of interest has several unknown and time-homogeneous
parameters, \(\theta \)  say, then one is interested to estimate such parameters, for instance using likelihood based methods. We show how our estimate can be leveraged for this latter task. In this paper we use the simultaneous stochastic perturbation stochastic approximation (SPSA) method [2]} popular in engineering applications. SPSA is based upon a finite difference estimator of the gradient, w.r.t. \(\theta \) , of the log-normalizing constant. It constructs a stochastic approximation scheme for the estimation of static parameters. We are not aware of this method being used for the EnKBF.
As it is a zero-order optimization method, we expect to be computationally less expensive than resorting to using other estimates of the gradient (of the log-normalizing constant).
It should be noted that our work focusses upon the standard or `vanilla' EnKBF, the deterministic EnKBF [3]} and deterministic transport type EnKBFs [4]};
other extensions are possible, for instance, to the feedback particle filter.
